$$f(x)=\int_{-\infty}^{\infty}\frac{e^{-a(x+z)^2}J_{1}(p\sqrt{b^2-z^2})}{\sqrt{b^2-z^2}}dz$$
I checked for integrals of the above mentioned form in Gradshetyn and Ryzhik, table of integrals, series and products and Luke's integrals of Bessel functions but couldn't find any. However, I was able to evaluate the integral using Mathematica's NIntegrate and plotted the answer as a function of $x$ but I am seeking to obtain the explicit form of $f(x)$. A guidance on how to go about evaluating this would be appreciated.